Normal Distribution — nothing is called 95% until it has been counted Illustrated essays on probability and statistical inference with the properties measured rather than asserted: every interval's coverage is counted, every test's p-values are checked for uniformity, every simulation is seeded, and the famous methods that fail their own claims are shown failing. https://www.normaldistribution.xyz/ 2026-08-06T10:58:52.221Z What the 95% refers to https://www.normaldistribution.xyz/essays/what-95-percent-means/ 2026-08-06T10:58:52.221Z An interval that claims 95% is making a checkable statement about a procedure, not about the interval in front of you. Build every possible sample and count, and the interval taught first turns out to cover 87.6% of the time. Sums of almost anything https://www.normaldistribution.xyz/essays/sums-of-anything/ 2026-08-06T10:58:52.221Z The theorem says sums converge on one shape whatever they are sums of, which is remarkable and true. Watching it happen from a one-sided skewed source, with the rate of convergence predicted in advance, is more convincing than watching the shape appear. A p-value that is not flat is not a p-value https://www.normaldistribution.xyz/essays/p-values-must-be-flat/ 2026-08-06T10:58:52.221Z Under a true null, p-values are uniform. That is stronger than saying the test rejects 5% of the time, it constrains the whole distribution rather than one point of it, and it catches implementation errors that a rejection rate sails past. Simpson's reversal is a region, not a table https://www.normaldistribution.xyz/essays/simpsons-reversal/ 2026-08-06T10:58:52.221Z The treatment wins in both groups and loses overall. That is normally shown with one famous table, which cannot answer the two questions a reader has — how often, and how large. Swept, it turns out to occupy 31% of the allocation space. The seed is part of the figure https://www.normaldistribution.xyz/essays/the-seed-is-part-of-the-figure/ 2026-08-06T10:58:52.221Z Every other site in this fleet draws from a deterministic rule, so a figure either is or is not what it claims. Here the figures are samples, and a sample can be right by luck. That changes what a figure has to carry. More data is not monotonically better https://www.normaldistribution.xyz/essays/more-data-is-not-monotone/ 2026-08-06T10:58:52.221Z Coverage of an interval for a proportion does not improve smoothly as the sample grows. It oscillates, and there are larger samples that cover materially worse than smaller ones — a sample of twenty covers twelve points worse than a sample of nineteen. The tail converges last https://www.normaldistribution.xyz/essays/the-tail-converges-last/ 2026-08-06T10:58:52.221Z The central limit theorem is usually shown as a shape arriving. What the demonstration leaves out is the rate — and the rate is wildly different in the middle and in the tail, which is where every approximation in the subject is actually read. What a p-value does not say https://www.normaldistribution.xyz/essays/what-a-p-value-does-not-say/ 2026-08-06T10:58:52.221Z The same p of 0.04 corresponds to a large effect in ten observations and a negligible one in two thousand. A p-value alone cannot be interpreted, and the number that makes it interpretable is almost never printed beside it. What a positive test is worth https://www.normaldistribution.xyz/essays/what-a-positive-test-is-worth/ 2026-08-06T10:58:52.221Z A test that is 90% sensitive and 95% specific sounds accurate. For a condition affecting one person in a thousand, 98% of its positive results are wrong, and a worse test on a commoner condition beats a better test on a rare one. Two routes to every number https://www.normaldistribution.xyz/essays/two-routes-to-every-number/ 2026-08-06T10:58:52.221Z A site about probability that only simulates has one route to each answer and no way to tell a right one from a plausible one. Every important number here is computed twice, by arithmetic that shares nothing, and the two are required to agree. What normal actually looks like https://www.normaldistribution.xyz/essays/what-normal-actually-looks-like/ 2026-08-06T10:58:52.221Z A single quantile plot of forty normal points wanders enough to look suspicious. Twenty of them, all genuinely normal, show what the noise looks like — and any single panel a reader would have rejected is in there. Twenty intervals and one expected miss https://www.normaldistribution.xyz/essays/twenty-intervals-and-one-miss/ 2026-08-06T10:58:52.221Z The 95% belongs to the procedure, not to the interval in front of you. Twenty intervals from twenty samples make that visible in a way no definition does, and the one that misses is not a mistake. The winner's curse https://www.normaldistribution.xyz/essays/the-winners-curse/ 2026-08-06T10:58:52.221Z Filter honest studies down to the ones that reached significance and the effects they report are systematically too large. At low power the inflation is a factor of two, nobody has done anything wrong, and the selection did all of it. Regression to the mean https://www.normaldistribution.xyz/essays/regression-to-the-mean/ 2026-08-06T10:58:52.221Z Select the worst performers, measure them again, and they improve. Select the best and they decline. No intervention is required for either, the size of the apparent effect is predictable from the correlation alone, and it is the reason so many things appear to work. The shape, and where its mass is https://www.normaldistribution.xyz/essays/the-shape-and-its-mass/ 2026-08-06T10:58:52.221Z 68, 95, 99.7 is recited more often than any other set of numbers in the subject. They are integrals of a specific curve, they are worth computing rather than remembering, and the third one is the one people misuse. The shortest interval is the one that misses https://www.normaldistribution.xyz/essays/the-shortest-interval-is-the-one-that-misses/ 2026-08-06T10:58:52.221Z Four intervals for the same data, with their widths and their coverage measured together. The narrowest is the one that fails its stated level, which is exactly why it looks the most appealing. Twenty analyses of nothing https://www.normaldistribution.xyz/essays/twenty-analyses-of-nothing/ 2026-08-06T10:58:52.221Z Twenty honest, correct analyses of data with no effect in it find something significant 58% of the time. Nobody p-hacked, every individual p-value is right, and the reported one is the smallest of twenty. Where the bootstrap lies https://www.normaldistribution.xyz/essays/where-the-bootstrap-lies/ 2026-08-06T10:58:52.221Z Resampling is the most generally useful trick in the subject and it has a failure mode that is easy to state: it cannot see past the data. For a statistic that lives at the edge of the sample, coverage collapses from 95% to almost nothing. The correction for not knowing the spread https://www.normaldistribution.xyz/essays/the-t-that-fixes-a-small-sample/ 2026-08-06T10:58:52.221Z The t distribution exists because the standard deviation is estimated rather than known. At eight observations, using the normal instead makes every interval 12% too short — and the coverage that follows can be measured rather than argued about.